Quasivariety

In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class.

Definition

A trivial algebra contains just one element. A quasivariety is a class K of algebras with a specified signature satisfying any of the following equivalent conditions:

1. K is a pseudoelementary class closed under subalgebras and direct products.
2. K is the class of all models of a set of quasi-identities, that is, implications of the form $s_1\approx t_1\wedge \dots \wedge s_n\approx t_n\to s\approx t$, where $s,s_1,\dots ,s_n,t,t_1,\dots ,t_n$ are terms built up from variables using the operation symbols of the specified signature.
3. K contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products.
4. K contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproducts.

Examples

Every variety is a quasivariety by virtue of an equation being a quasi-identity for which Template:Nowrap.

The cancellative semigroups form a quasivariety.

Let K be a quasivariety. Then the class of orderable algebras from K forms a quasivariety, since the preservation-of-order axioms are Horn clauses.